Analysing English year-one mathematics textbooks through the lens of foundational number sense: A cautionary tale for importers of overseas-authored materials

ABSTRACT In this paper we present analyses of three textbooks currently used in the teaching of mathematics to year-one children in England. One is an established English-authored textbook, while the others are Singaporean-authored imports promoted by government as solutions to perceptions of systemic failure. Every task in each textbook was coded against a set of eight number-related competences known to support children’s learning in both short and long terms. Such a framework, which is literature-derived and curriculum-independent, enables meaningful comparison of materials deriving from different cultural contexts. Analyses of the proportions of all tasks coded for the different competences showed that none of the three books adequately addresses all eight competences, although the English-authored comes closest. Moving averages, undertaken to show the temporal location of the opportunities presented for children to acquire the eight competences, showed them distributed throughout the school year in the English-authored textbook but only during the first half of the school year in the two Singaporean-authored textbooks. Some implications for the importation of such materials are discussed.


Introduction
Every aspect of school mathematics, including the production and deployment of textbooks (Sayers et al., 2021), is underpinned by cultural norms and expectations (Andrews, 2016). In Cyprus, for example, teachers have no option but to use government-produced textbooks (Xenofontos, 2019). In Singapore, commercially produced textbooks require ministerial approval before use (Kaur, 2014). In England, the site of the research presented here, the production of textbooks is effectively unregulated (Jones & Fujita, 2013), although, as we discuss below, a somewhat shadowy committee has been appointed to approve textbooks for a government-funded scheme designed to fulfil an education minister's desire to import Singaporean mathematics teaching into English schools (Ward, 2018). That said, in all three instances, albeit with differing levels of regulatory confidence, textbooks mediate the intentions of a centralised curriculum and teachers' practices (Valverde et al., 2002).
Differences in the regulation of textbook production make different professional demands on teachers. In Cyprus, where the textbook is the curriculum, teachers have no need to evaluate textbooks' curricular resonance. In Singapore, confident that textbooks' curricular coverage has passed official scrutiny, teachers may need to evaluate differences in emphasis or the sequencing of mathematical ideas to decide which books best suit their needs. However, English teachers need to be able to evaluate both textbooks' curricular resonance and their sequencing of mathematical content. This latter matter has been made more pertinent by the introduction, following extensive ministerial encouragement, of Singaporean-authored textbooks into English classrooms. Moreover, providing an oblique motivation for this paper, a recent analysis of Swedish year-one children's textbooks, authored by Swedes, Finns and Singaporeans respectively, found the latter to be the least well-suited of the three books to facilitate children's acquisition of a set of eight literature-derived and curriculum-independent number competences that underpin later learning (Sayers et al., 2021).
In this paper, exploiting the same analytical framework as in the Swedish study, we analyse three textbook series currently used in English year-one classrooms. Two of these are Singaporean imports, while the third is a well-established and popular Englishauthored series. With respect to the latter, our view is that a textbook's longevity provides an indication of its curricular resonance and appropriateness as a benchmark for comparison. But first, we examine the warrant for the importation of Singaporean textbooks and what evaluative evidence has said with respect to their adequacy and relevance in cultural contexts other than the one for which they were written. Following this, we introduce our analytical framework and present the warrant for our research question.

On the international importation of Singaporean mathematics textbooks
For the last quarter century, Singapore's consistently high performance on two international large-scale assessments (ILSAs) of students' mathematics competence, the Trends in International Mathematics and Science Study 1 (TIMSS) and the Programme of International Student Assessment 2 (PISA), has created much international interest in its school textbooks. For example, Singaporean mathematics textbooks were imported into the United States to address perceptions of American student underachievement (Beckmann, 2004;Hoven & Garelick, 2007). Evaluations of these textbooks, known as Singapore Maths, yielded ambivalent results. On the one hand, the US government's own syntheses of available research found no evidence of any significant impact of Singapore mathematics on either primary or secondary students' achievement (United States, Department of Education, 2015). On the other hand, a government-sponsored intervention found that schools with committed teachers and stable populations of high-performing students produced sizeable improvements in student learning, while schools with uncertain teacher commitment or transient lower income populations produced uneven or disappointing results (Ginsburg et al., 2005). Analyses of a later version of the same textbook series concluded that its approach to the teaching of the equals sign, which privileged a procedural rather than a relational understanding, had the potential to compromise children's later learning of algebra (Powell, 2012). Elsewhere, the same series has been introduced into Sweden as Singma, where comparisons with a popular Swedish-authored textbook and a Finnish import found, as indicated above, that Singma was the least adequate curriculum match of all three books and, importantly, a potential disruption to traditional classroom practices (Sayers et al., 2021).
A second Singapore textbook series, 'My Pals are Here' (hereafter, MPAH), has been widely adopted internationally and prompted researchers to compare, in relation to other countries' home-produced textbooks, how MPAH different topics are approached. For example, a comparison of MPAH and various US textbooks found, with respect to the teaching of length measurement, that all textbooks were procedurally dense and addressed conceptual knowledge in similar proportions (Lee & Smith, 2011). The same study found MPAH comprising many more worked examples than its US equivalents, although the US textbooks paid more attention to the use of non-standard units, known to be a predictor of later learning (Chang et al., 2011;Desli & Giakoumi, 2017). Another study compared geometry-related tasks across primary textbooks from five countries, although here we discuss only MPAH and its US equivalent. Here, Wang and Yang (2016) found MPAH comprising three times as many geometry-related tasks as the US book. However, while MPAH comprised a much higher proportion of symbolically presented tasks, the US book comprised a much higher proportion of verbally presented tasks. Moreover, while there were no differences in the proportions of contextual and non-contextual tasks, almost all tasks in MPAH were closed, compared to almost a third in the US book being open.
Elsewhere, a comparison of algebra-related tasks in primary textbooks from Finland, Singapore and Taiwan found MPAH comprising a higher proportion of contextual tasks than the Finnish textbook but a significantly lower proportion than the Taiwanese. Also, while both MPAH and the Taiwanese textbook comprised equal proportions of symbolically and verbally presented tasks, the Finnish book comprised a significantly higher proportion of the former and a significantly lower proportion of the latter (Yang & Chiang, 2017). Recently, MPAH has been introduced into the Netherlands, principally to 'enhance students' problem-solving skills' (Van Zanten & Van den Heuvel-Panhuizen, 2018, p. 830). Analyses indicated that MPAH offered comparable proportions of routine tasks to the Dutch-authored textbooks under scrutiny. That said, MPAH 'systematically offered directions for both students and teachers that can be interpreted as facilitators for the learning of problem solving' that was missing in the other books, although the nature of these support materials tended to compromise the authenticity of the included problems (Van Zanten & Van den Heuvel-Panhuizen, 2018, p. 835).
Overall, our interpretation is that such differences, whether didactical or contentrelated, should not of themselves be construed as strengths or weaknesses but indications of differing cultural norms and expectations with respect to what mathematics is privileged locally. Such differences, therefore, may make the uncritical importation of MPAH problematic.
English interest in Singapore's textbooks has been similarly driven by its ILSA-related successes. For example, in 2014, an education minister, implying the same could not be said of home-produced materials, praised Singapore's secondary textbooks' 'clear structure, strong explanations of key ideas, helpful worked examples and plenty of opportunity for essential practice to increase fluency and understanding'. 3 Two years later, the same minister asserted that an enormous amount of thought and care goes into the construction of (Singapore's) maths textbooks, planning in great detail the sequence of teacher exposition. No pupil's understanding is left to chance or accident: every step of a lesson is deliberate, purposeful and precise. 4 By 2017, the minister announced the decision to fund 'southeast Asian approaches to maths teaching', 5 including the use of maths textbooks from Singapore. 6 This decision, offering both rhetorical and financial encouragement for the importation of Singaporean textbooks, prompted the creation of an 'expert panel' charged with evaluating materials submitted by publishers seeking official approval to participate in the funding scheme. Membership of this 'expert panel' was so secretive, it took a freedom of information request to the ministry to identify it (Ward, 2018).
The totality of the above confirms the cultural uniqueness of school mathematics (Andrews, 2016). It suggests that the minister's largely uncritical promotion of Singaporean textbooks may be problematic (Sayers et al., 2021;Van Zanten & Van den Heuvel-Panhuizen, 2018), prompting the question: 'Will insisting that all primary teachers teach from textbooks, all of which must conform to the format, style and pedagogy of the Chinese or Singaporean ones, actually produce the results the minister presumably desires?' (Merttens, 2015, p. 391).
However, before presenting our analytical framework, it seems reasonable to ask whether international interest in Singapore's mathematics achievement is as warranted as the ILSA data suggest. In this respect, we turn to Singapore's publicly available national data. The Department of Statistics' figures for 2019 show 1,677,400 people, 29.4% of the population, living on the island but not classed as resident. 7 Singapore's Department of Manpower's figures show that 1,399,600 of these 8 were foreign employees who do not satisfy the criteria for permanent residence. Of these, 981,000, were 'work permit' employees 9 who are not permitted to have families join them. The majority of the remainder, representing two different employment categories, are permitted to have family join them on condition that their monthly salaries exceed SGD 6,000. However, as the monthly threshold salaries for these two groups are SGD 3,600 10 and SGD 2,400 11 respectively, it seems unlikely that workers in either group, 189,000 in the first and 197,800 in the second, earn sufficient for their families to join them. In short, Singapore's own figures suggest that between one and 1.4 million workers will not have family permitted to live with them. Finally, of the resident population, 2,628,800 people fall into the age range 20-64 and can be construed as representing, assuming that all people in this age range are employed, the working resident population. This figure is roughly twice the number of foreign workers, few of whom will have family living with them. Thus, while such calculations are necessarily approximate, it seems reasonable to infer that the children of at least a third of the working population of Singapore are not permitted to live on the Island, with the consequence that when ILSAs are conducted, the children of the lowest-paid workers are systematically excluded. We return to this matter below.

A framework for analysing essential number-related competences
The analyses below exploit the eight categories of foundational number sense (FoNS; Andrews & Sayers, 2015). FoNS is the outcome of constant comparison analyses (Charmaz, 2008), undertaken to identify the number-related competences all year-one children should acquire if they are to become successful learners of mathematics, of international literature drawn from mathematics education, psychology, special educational needs and generic education. FoNS bridges the innate competences of the approximate number system (Dehaene, 1997) and the number-related competences 'required by all adults regardless of their occupation' (McIntosh et al., 1992, p. 3). The eight categories of FoNS, each requiring instruction, offer 'a simple to operationalise framework . . . with the potential to inform teacher education, facilitate classroom evaluations and provide a warranted tool for use in cross-cultural studies' (Andrews & Sayers, 2015, p. 258). Earlier frameworks (see, for example, Howell & Kemp, 2005;Purpura & Lonigan, 2013) comprised too many categories for use in the contexts for which FoNS, summarised in Table 1, was developed. Hitherto, the FoNS framework has successfully, and without cultural bias, facilitated analyses of Swedish year-one textbooks, including Finnish and Singaporean imports (Sayers et al., 2021), and number-related learning opportunities of year-one classrooms in England, Hungary and Sweden (Andrews & Sayers, 2015;Sayers et al., 2016).

Methods
Methodologically, and acknowledging potential incompleteness (Rezat, 2006), classroom-independent analyses can foreground both curricular resonance and curricular omissions (Yang & Lin, 2015). They can foreground the influence of authors' personal perspectives on the development of mathematical competence, and whether those perspectives are informed by research (Hodgen et al., 2010). Such analyses matter because curricula, and the textbooks that reify them, inform 'what students learn, when they learn it, and how well they learn it' (Cai & Cirillo, 2014, p. 133). In the following, we present analyses of the FoNS-related learning opportunities found in three year-one mathematics textbooks, two Singaporean-authored and one English, framed by the question: How do adaptations of two Singaporean textbooks and a popular English-authored textbook structure opportunities for English year-one children to acquire the eight competences of foundational number sense? Because textbook production in England is effectively unregulated (Jones & Fujita, 2013), the two Singaporean imports have been compared with an established English-authored series, whose popularity implies an acceptable curricular resonance and an indication of long-standing pedagogical norms and expectations. According to their publishers' websites, all three books address current curricular expectations, particularly with respect to children's mastery of mathematics, a label promoted for what the international literature would describe as good practice. Abacus, which can be construed as typical of Englishauthored textbooks, aims to achieve mastery of the key concepts and skills of school mathematics. 12 Inspire Maths (IM) is an adaptation of MPAH. An initial evaluation of its impact on English children's achievement found a small but significant impact on learning (Lindorff et al., 2019), prompting its publisher to claim, somewhat disingenuously, that it is (our emphasis) the 'only mastery textbook programme proven to raise standards in the UK'. 13 The origins of Maths -No Problem (MNP), the second Singapore-based scheme under scrutiny, seem opaque. Boylan (2021, p. 16) writes, without elaboration, that it was 'based on a translation of a Singaporean textbook series', 'initiated by entrepreneurs who had previously lived in Singapore'. Its publisher asserts that 'it is recommended by the DfE for schools on the Teaching for Mastery Programme and is fully aligned with the 2014 English national curriculum'. 14 However, evaluations of its efficacy seem scant, as researchers seem to have focused more on its impact on practice rather than achievement (see, for example, Blausten et al., 2020;Boyd & Ash, 2018;Unsworth & Tummons, 2021). Because most FoNS categories represent generic forms of competence, FoNS-related tasks are likely to be distributed throughout the various chapters or sections of the textbooks under scrutiny. Such circumstances demand particular attention to the unit of analysis (Stylianides, 2014). In this respect, a recent compilation, typically focused on the presentation of reasoning and proof (RP) in various US school textbooks, offered helpful insights. One study coded 'every other lesson or investigation in each unit or chapter' (Bieda et al., 2014, p. 74). Another 'sampled roughly half of the sections from each textbook' (Davis et al., 2014, p. 99), while a third analysed every task in chapters designated RP and a random selection of all others (Otten et al., 2014). A fourth focused, somewhat vaguely, 'on locations in textbooks where authors explicitly addressed issues related to reasoning-and-proving', seeking 'reasoning-and-proving in the textbook without reading every chapter and section' (McCrory & Stylianides, 2014, p. 120). Overall, these different sampling approaches, alongside the application of, effectively, the same analytical framework, led the various authors to report only frequencies and proportions, although one study also included chi-square analyses of the proportional distribution of the various codes to highlight significant differences in emphasis (Davis et al., 2014). Such procedures, as conceded by Bieda et al. (2014), fail to account for any RP activities outside the sampled material and, importantly in the context of a study focused on textbooks' structure, the temporal location of such material within the overall timescale of the textbook. In the context of FoNS, and to avoid problems associated with sampling, the decision was made to analyse all tasks in the books under scrutiny according to the following schedule.
Two colleagues independently analysed all tasks in each book that expected some sort of response from the child and recorded any opportunity for the development of any FoNS category. In addition, to provide an understanding of the broader context in which FoNS-related opportunities were located, colleagues coded any number-related tasks falling outside the FoNS framework as FoNS+ and any tasks addressing mathematical topics unrelated to number as other. However, as the explicit focus was on FoNS, no further attention was paid to that nature of any FoNS+ or other tasks. On completion of these independent coding processes, the project team met to discuss and resolve the coding of any tasks on which the original coders disagreed. Thus, the codes assigned to each task were either agreed by two independent coders or four colleagues discussing them collectively. Importantly, most tasks yielded more than one code. For example, Figure 1 shows two tasks found in a popular English textbook. Their contexts within the book indicated that one had an explicit focus on simple arithmetical operations, while the other addressed number patterns. However, while their accompanying text offered no indication of any additional expectations, the project team construed both as offering additional learning opportunities. The left-hand task was additionally coded for number recognition and, due to its allusion to cardinality, awareness of the relationship between number and quantity. The right-hand task was additionally coded for number recognition and, as both were viable solution strategies, systematic counting and simple arithmetical operations. This approach to coding, involving colleagues raised and educated in three different European cultures, avoided the problems of mono-cultural lenses and ensured that all tasks were examined for any potential FoNS-related learning. That is, the process was designed to uncover, in a manner that was as value-free as possible, all FoNS-related opportunities, irrespective of authors' expressed intentions.
Two analyses were undertaken. The first, motivated by substantial differences in frequencies, involved analyses of the proportions of all tasks identified as addressing the different FoNS and other competences. Such an approach facilitates more meaningful comparisons than simply comparing frequencies. The second involved the calculation of moving averages to show how a textbook sequences mathematical content over time. The decision to use moving averages was motivated by the Swedish study discussed above (Sayers et al., 2021), which highlighted their efficacy with respect to exposing textbooks' structural emphases. In this study, calculations were based on an interval of one week, which entailed dividing the total number of tasks in the whole book by 40, the number of weeks in the school year, to obtain an estimate of each week's workload. Of course, when using textbooks, teachers may choose to teach some elements out of sequence and few children will undertake every task. However, assuming that authors structure textbooks according to some principle of mathematical development, we have construed textbooks as ordered sequences of activity. Drawing on this assumption, moving averages highlight authors' sequencing decisions, making them a powerful comparative tool. With respect to interpretation, whenever a graph shows above zero any student who has completed all possible tasks would have met that coded property during the week in question. Also, the higher the graph the more frequently the code will have been observed, to the extent that when a graph reaches a height of one, every task included in that week's calculation will have addressed that particular property. Finally, a horizontal line at height one means that every task over the period represented by that line addresses the particular characteristic under scrutiny.

Results
In the following we present the two analyses of the three books. Table 2, in addition to the total number of tasks in each book, shows the frequency and the proportion of all tasks in each book coded for FoNS and other competences. It is an interesting coincidence that the total number of tasks in Abacus (1524) is 78% of the total found in MNP (1955), which, in turn, comprises 78% of the total found in IM (2494).
An initial scrutiny of the figures of Table 2 confirms the problematic nature of frequency-based analyses. For example, the number of tasks coded for number recognition and the relationship between number and quantity appear remarkably similar across the three books, potentially warranting a conclusion of similarity of learning opportunity, a conclusion substantially challenged when such figures are viewed proportionally. Moreover, assuming a school year of 40 weeks, one conclusion might be that Abacus, with 38 tasks per week, has lower expectations than MNP, 49 tasks per week, and IM, with 62. However, assuming that the number of codes simultaneously applied to a task is an indicator of task complexity, Abacus, with 3.0 codes per FoNS-coded task, could be construed as marginally more complex than MNP, 2.8 codes, and IM, 2.6 codes. In other words, there are indications that as the number of tasks increases the incidence of simultaneous codes, and therefore task complexity, reduces. However, the focus of this study is not task complexity, which would require an entirely different analytical framework, but the extent and location of FoNS-related opportunities. To that end, we argue that analyses based on proportionality present a clearer picture of children's opportunities to learn, particularly as all three books have been promoted by their publishers as addressing the same curricular expectations. In other words, showing one textbook addressing particular forms of learning in ten percent of all tasks and another addressing the same learning in twenty percent of all tasks is a transparent uncovering of the books' differing emphases. With respect to the particular FoNS competences, and drawing on Davis et al. (2014) above, the figures of Table 2 have been supplemented by chi-square tests, shown in Table 3, which examined the proportional distribution of the various codes between the different pairs of books. From the perspective of similarities, the figures of Table 2 show only three FoNS competences are addressed in comparable proportions across the three books. These are systematic counting, which occurs in between 14% and 16% of all tasks, quantity discrimination, which is collectively addressed in low proportions, between 4% and 6% of all tasks, and estimation, which occurs so infrequently that chi squares could not be calculated. The chi square tests, where calculable, confirmed this sense of similarity.
The figures of Table 2 show MNP comprising a higher proportion of tasks than IM for each of the six FoNS competences addressed by both books, although the chi square tests suggest that none of these differences is significant. The only deviation from this pattern concerns number patterns, which is absent in MNP and minimally present in IM. Overall, the chi square tests, where calculated, show no significant differences between the two Singaporean books on any analysed characteristic.
Abacus, by way of comparison with the two imports, comprises a significantly higher proportion of tasks coded for number recognition than IM but not, although the probability hints at such a difference, MNP. Abacus also comprises a significantly higher proportion of tasks coded for simple arithmetical operations than either IM or MNP. Also, although chi squares cannot be calculated, it addresses number patterns in ways that distinguish it from both IM and MNP, and, albeit in miniscule amounts, estimation. Alternatively, there are no FoNS competences for which either IM or MNP has a significantly higher proportion of tasks than Abacus.
Moving beyond the FoNS competences, the figures of Table 2 show Abacus comprising (50%) a higher proportion of tasks coded for at least one FoNS category than either MNP (41%) or IM (36%), although, as shown in Table 3, the difference is significant only with respect to the comparison between Abacus and IM. Table 2 also shows IM comprising a higher proportion of tasks coded for FoNS+ than either Abacus or MNP. Indeed, the difference between IM (43%) and MNP (30%) is close to being the only significant difference between the two Singaporean books. By way of contrast, MNP comprises a higher proportion of other tasks than IM, which comprises a higher proportion than Abacus. Overall, however, despite the relatively few significant differences, it seems reasonable to infer that the two Singaporean imports comprise not dissimilar proportions of all categories of learning opportunities, whether FoNS or otherwise. It also seems reasonable to infer that, with a few notable exceptions, the two Singaporean imports offer comparable proportions of opportunity to Abacus. This apparent similarity prompts the second analysis. Figure 2 shows the moving averages for the five FoNS competences that attracted sufficient codes to merit constructing the graphs. The graphs, where a solid line represents IM, a broken/dashed line represents Abacus and a dotted/greyline represents MNP, distinguish Abacus, on the one hand, from IM and MNP on the other. In all instances, it can be seen that beyond the midpoint of the school year, neither IM nor MNP offers any opportunities for students to engage with number recognition, systematic counting, number and quantity, different representations of number or simple arithmetical operations. By way of contrast, Abacus continues to offer opportunities for all five competences, albeit in different intensities, throughout the second half of the school year.
Finally, returning to those tasks not coded for any FoNS competence, the graphs of Figure 3 further highlight structural similarities and differences. Here, the second half of IM's school year is dominated by FoNS+ tasks, with only a small dip for tasks focused on other topics. By way of contrast, MNP's second half begins with an intensive phase whereby every task for two months relates to FoNS+, before being replaced by an emphasis on other topics. What is most interesting, however, is the absolute exclusivity of the two forms of task in the two books, as confirmed by the continuous horizontal lines of the combined graphs, showing that every task in the second half of both books is either FoNS+ or other but never both. Abacus, on the other hand, has neither dominant periods nor lengthy periods of such task-type exclusivity. Such matters allude to structural differences between the differently-authored books. On the one hand, Abacus seems to introduce FoNS+ and other at low levels early in the year and gradually increases their intensity alongside continuing opportunities for FoNS. On the other hand, both IM and MNP effectively distinguish between FoNS and FoNS+ at the middle of the school year, after which both have distinct sections addressing FoNS+ and other.

Discussion
In this paper, acknowledging that the lack of textbook regulation in England (Jones & Fujita, 2013) necessitates teachers being able evaluate textbooks' curricular resonance and sequencing of mathematical content, and motivated by the introduction of Singaporean imports into English year-one classrooms, we have analysed the opportunities found in three mathematics textbooks, one English-authored and two Singaporeanauthored, for children to acquire the eight research-derived and curriculum-independent competences of foundational number sense (FoNS), each of which is implicated in children's later learning of mathematics (Andrews & Sayers, 2015). The results show both similarities and differences in how mathematics generally and FoNS-related opportunities are presented. Importantly, despite earlier concerns (Rezat, 2006), the results demonstrate the value of classroom-independent analyses.
With respect to differences, the total number of tasks in each of the three books varies considerably, with the English-authored Abacus comprising 78% of the total found in MNP, which, in turn, comprises 78% of the total found in IM. This variation implies differing expectations with respect to the number of tasks to be completed in any given time-period, warranting an initial conclusion that IM implies a greater workload than MNP, which, in turn, implies a greater workload than Abacus. However, this may be too simplistic a picture. On the one hand, the generally higher proportion of FoNS-related tasks found in Abacus, alongside evidence that FoNS-related tasks are likely to be more complex in Abacus than either IM or MNP, suggest that Abacus may offer a stronger foundation for children' learning than either of the Singaporean imports. On the other hand, the higher proportions of tasks coded for both FoNS+ and other in both IM and MNP suggest that the Singaporean imports may expect children to engage with more difficult material more frequently and, supported by the structural evidence of the moving averages and earlier comparative studies involving Singaporean textbooks (Cai et al., 2005), more quickly than Abacus. Thus, a not unreasonable conclusion would be that the Singaporean books may be better suited to highly motivated and high-achieving students with well-qualified and committed teachers, as found in an earlier American study (Ginsburg et al., 2005), an inference that accords with the apparent lack of low paid workers' children in Singapore's schools. In short, the structures of the two Singaporeauthored textbooks, as highlighted by the recent Swedish study, may disrupt traditional classroom expectations and practices (Sayers et al., 2021). Moreover, without further analyses focused on a systematic evaluation of task complexity, workload, particularly when located in an analysis of frequencies, remains a problematic construct.
From the perspective of the individual FoNS competences, the proportions of the books given over to number recognition, simple arithmetical operations and number patterns vary considerably across the three books. All three categories, which occur in higher proportions in Abacus than either IM or MNP, have been implicated in the later learning of mathematics. For example, the positive impact of number recognition on later achievement, particularly multi-digit arithmetic (Desoete et al., 2012), lasts as late as adolescence (Geary, 2013). Similarly, simple arithmetical competence predicts mathematical learning in general (Krajewski & Schneider, 2009) and problem solving in particular (Levine et al., 1992). Further, the limited number patterns-related opportunities found in both IM and MNP may have unforeseen long-term implications, as a lack of familiarity with number patterns is an important predictor of later mathematical difficulties (Lembke & Foegen, 2009), while recognising, extending and repeating patterns is a prerequisite of mathematical success (Wijns et al., 2019). Moreover, acknowledging that all FoNS competences are dependent on instruction, it is important to note that early pattern-related interventions influence later achievement and narrow the gap between high-and low-achievers (Mulligan et al., 2013). Of course, it is possible that the lack of number patterns in IM and MNP reflects cultural norms concerning what mathematics is privileged in early learning. That said, with respect to number recognition, simple arithmetical operations and number patterns, Abacus may be better placed to provide children with a core subset of basic competences than either IM or MNP.
With respect to similarity, the most obvious, across all three books, is the small proportion of tasks focused on quantity discrimination and, effectively, the absence of estimation. Both categories have been implicated in the mathematics achievement and, importantly, identified as key determinants of later learning difficulties. For example, quantity discrimination is a strong predictor of general mathematical achievement (Desoete et al., 2012;Lipton & Spelke, 2005), while the ability to estimate predicts mathematics achievement in general (Libertus et al., 2013;Simms et al., 2016) and arithmetical competence in particular (Cornu et al., 2017;Dietrich et al., 2016). Moreover, the development of the ability to estimate is not a chance phenomenon but requires intervention (Praet & Desoete, 2014). In sum, when set against the core competences of the FoNS framework, all three books have omissions that may result in diminished mathematical understanding.
Moving beyond the FoNS framework, the proportions of number-related tasks falling outside the FoNS framework (FoNS+ and other) highlights further differences between the English-and the Singaporean-authored textbooks, both in terms of expectations and structure. From the perspective of expectations, Table 2 showed IM comprising a significantly higher proportion of FoNS+ tasks than MNP, with Abacus falling between the two. By way of contrast, MNP comprises a higher proportion of other tasks than IM, which comprises a higher proportion than Abacus. In other words, if Abacus is the benchmark against which English curricular resonance is evaluated, the Singaporean imports, while significantly separated themselves, offer reasonable approximations to the proportion of FoNS+ tasks found in Abacus. However, neither approximates the proportion of Abacus' FoNS-related tasks and only IM could be construed as comparable with regard to other. Thus, IM may be a better adaptation for English use than MNP.

Conclusions
The above prompts several conclusions with respect to the three textbooks' adequacy for facilitating year-one children's acquisition of FoNS and later learning. Firstly, none of the books addresses all eight FoNS categories adequately, although Abacus seems better placed than either IM or MNP. This lack is particularly evident with respect to estimation and, in the case of the two Singapore-authored books, number patterns. Secondly, any adoption of either IM or MNP, with their structural differences and rapid shifts to more difficult mathematics, is likely to challenge the competence of English primary teachers, not least because Singaporean teachers are typically specialists, while England's are typically generalists (Blausten et al., 2020;Cantley, 2019;Merttens, 2015). Moreover, the adoption of either IM or MNP will require professional development opportunities comparable to those offered to their Singaporean colleagues (Ginsburg et al., 2005). In this respect, we note that for more than twenty years every teacher in Singapore has an annual entitlement of 100 hours of funded professional development, which may take a variety of forms tailored to the individual (Leong et al., 2011). By way of contrast, the English publisher of MNP currently advertises a range of courses that would, assuming that one teacher from a typical primary school attended all relevant courses, subsume the whole of the government initiated financial inducement. In short, the textbook-related professional development opportunities for English teachers of year-one mathematics not only fall considerably short of those available to their Singaporean colleagues but are considerably more expensive.
The findings above allude to at least four hitherto unacknowledged issues. First, can textbooks developed for use in elite systems ever be appropriate in cultures that educate all children irrespective of family income? Alternatively, is the importation of Singaporean textbooks an example of education ministers promoting overseas practices in ways that ignore the social, cultural, economic, historical and political traditions in which those practices were originally located (Cantley, 2019)? Second, to what extent are the criteria 15 used by the members of the mysterious 'expert panel' informed by research or concerns that such materials should be fit-for-purpose with respect to the English curriculum? Third, to what extent have the English authorities sought solutions to perceived educational failures closer to home? For example, Flanders, the Dutch-speaking region of Belgium, has consistently been Europe's highest achiever on the mathematics components of both PISA and TIMSS (Andrews, 2015). Moreover, in becoming Europe's most successful education system, the Flemish authorities seem to have successfully addressed the education of a multi-ethnic community not dissimilar to that of England. Unfortunately, because PISA reports on Belgium as a whole, Flanders' achievements have gone largely unnoticed. Finally, highlighting the importance of understanding the role of population demographics on a nation's ILSA performance, what would be the impact on Singapore's ILSA-related mathematics scores were all its excluded children included? Alternatively, what would be the impact on England's ILSA scores if the children of the lowest-paid third of its workforce were excluded not only from ILSA participation but school itself? Notes Judy Sayers is an associate professor in mathematics education at the University of Leeds, UK. With extensive experience of mathematics teacher education at three universities in two countries, her research is typically focused on early years and primary mathematics education, with particular interests in making mathematics accessible to all learners through the improvement of mathematics teacher education. From 2016-2021 she was a core member of the Swedish Research Council-funded project team working on the development of foundational number sense in yearone children.
Eva Rosenqvist works in mathematics teacher education at Stockholm University, where her principal interests are the professional development of teachers of early years and primary mathematics. From 2017-2021 she was a research assistant for the Swedish Research Council-funded project on the development of foundational number sense in year-one children.
Paul Andrews is a professor of mathematics education at Stockholm University, Sweden and VIA University College, Aarhus, Denmark. His research typically involves comparative studies of all aspects of mathematics teaching and learning with the simultaneous goals of better understanding teachers' and students' mathematics-related beliefs, improving classroom practice, and challenging the uncritical use of the results of international tests of achievement as the basis for policy. From 2016-2021 he directed the Swedish Research Council-funded project on the development of foundational number sense in year-one children.